Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C22×C4 |
| direct product | C13×#32625 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C13 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C13 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C2×C4×Q8×C13 ◁ G | |
674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 3328 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,20)(16,19,17,21)(23,26)(24,28)(30,31)(32,35)(33,36)(34,37)(38,45)(39,43)(40,42)(41,44);(14,15)(16,19)(17,21)(18,20)(22,23)(24,27)(25,26)(28,29)(30,32,33,39)(31,35,36,43)(34,38,41,40)(37,45,44,42);(14,16,18,17)(15,19,20,21)(22,24,25,28)(23,27,26,29)(31,36)(32,38)(34,41)(35,42)(39,40)(43,45);(14,17,18,16)(15,19,20,21)(23,26)(27,29)520 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 25613);